General Numbers: The Number Plane Beyond the Right Angle

Each enlargement of the number system has answered a question that the old numbers could not. The last one in the plane, Wessel's geometric reading of the complex numbers, rests on a right angle: multiplication by i is a quarter turn, and the imaginary axis stands perpendicular to the real one. The General Sinus, a trigonometry of two variables defined on the oblique triangle, has freed sine and cosine from the right angle. This paper frees the frame in which the number plane is read. A General Number is z = a + b G, where the Composer Unit G is the arrow at the angle φ̄ to the real axis whose length is fixed, by one axiom, at |cos φ̄ + sin φ̄|. For each angle the General Numbers form a relative plane, and the complex plane is the relative plane φ̄ = π/2. Every true relative plane is the complex plane named in an oblique frame, and we give the isomorphism explicitly. In this frame we derive the Rule of the Square, G² = 2|G| cos φ̄ G − |G|², the conjugate, division and a Turning Unit J with J² = −1, and we write Euler's and De Moivre's formulas and the n-th roots with the General Sinus. The algebra of each relative plane is that of the elliptic generalized complex numbers; what is new is the frame fixed by the Axiom, its native trigonometry, and the laws of arithmetic written in it. We also extend the product formulas of the General Sinus Transform to any number of factors. All results have been verified symbolically and numerically.